So if I understand correctly the question you wanted to ask was:
Is it true that a triangle $$X \stackrel{u}{\to} Y \stackrel{v}{\to} Z \stackrel{w}{\to} \Sigma X$$ is split if and only if one of $u$, $v$, or $w$ is zero. The answer to this is yes.
It is clear (I think I can add details if someone wants) that if the triangle is split then one map must be zero (basically since we have an epi composing to zero). Conversely suppose that $w$ is zero, which is sufficient since we can always just rotate. Now we know by the axioms that $$Z \stackrel{-1}{\to} Z \to 0 \to \Sigma Z$$ and hence $$0 \to Z \stackrel{1}{\to} Z \to 0$$ are triangles (as an exercise check that any sequence of this form given by an isomorphism is necessarily a distinguished triangle), and $$X \stackrel{1}{\to} X \to 0 \to \Sigma X$$ is also a triangle. It is easy to check then that so is the direct sum $$ X \to X\oplus Z \to Z \stackrel{0}{\to} \Sigma X$$. The identity maps on $X$ and $Z$ then induce a map via [TR3] from this to the original triangle (since we have $w=0$ this trivially satisfies the necessary commutativity to apply [TR3]) and since two of the maps are isomorphisms so is the third. Hence any triangle as above with $w=0$ is isomorphic to one obtained by summing triangles on identity maps.
Proof of the claim that the direct sum of triangles is a triangle:
Let $X \to Y \to Z \to \Sigma X$ and $X' \to Y' \to Z' \to \Sigma X'$ be two distinguished triangles. We can complete the map $X\oplus X' \to Y \oplus Y'$ to a triangle $$X\oplus X' \to Y \oplus Y' \to Q \to \Sigma(X\oplus X')$$. We then get two diagrams whose rows are triangles by projecting to the factors $X, Y$ and $X', Y'$ respectively which we can complete to maps of triangles by the mapping axiom [TR3]. The maps $Q\to Z$ and $Q\to Z'$ such obtained induce a map $Q\to Z\oplus Z'$ by the universal property which gives a map from the triangle $X\oplus X' \to Y\oplus Y' \to Q$ to the pretriangle (a pretriangle is one where the maps compose to zero and it plays nicely with homological functors, that is they take it to a long exact sequence, this is clear from the fact that homological functors are additive and it is a sum of distinguished triangles) $X\oplus X' \to Y\oplus Y' \to Z\oplus Z'$. Two of the maps are isomorphims, namely the identities on the terms $X\oplus X'$ and $Y\oplus Y'$ so that the third must be also - this follows from the fact that $Hom(A,-)$ is a homological functor for any $A$, Yoneda, and the 5 lemma. So the triangle in question is isomorphic to a distinguished triangle and hence itself distinguished.
This works in the generality of arbitrary coproducts/products provided the coproducts/products in question exist and we consider a pretriangle to be one which homological/cohomological functors preserving the coproducts/products take to long exact sequences.
I'd recommend reading through the first chapter of Neeman's book Triangulated Categories - this is certainly covered in there as well as a bunch of other facts you might find useful in reading that paper. The reference for this result is Corollary 1.2.7 (it seemed lazy not to check since it is on my shelf) and the proof there is pretty much identical to the one here except that the facts I glossed over are proved earlier.
I'm not too familiar with Expositiones Mathematicae, but have you given them a look?
EDIT: The article I happened to have seen, which made me think that Expo Math might be along the lines Pete Clark was looking for, is this paper of T. Bühler - it modestly claims to no originality save for assembling disparate parts of the literature and writing down what's old news to connoisseurs (I'm paraphrasing here!) but of course this is, in a sense, precisely its originality & worth.
Best Answer
As this may be of use to others as well, I will try to provide some general points on journal selection for an `average' paper.
One obvious general approach to take is (i) to look where the references of your paper were published (and I note that there are lots still unpublished so here that may raise a problem... so check on whether they have now been published).
(ii) Look at, or estimate, the publication backlogs where available and make your decision on those grounds. (You have a delicate balancing act ahead as your idea of a two month acceptance is really hard to achieve and may need revising!)
(iii) Look at other journals similar to those found by (i), e.g. by iterating (i) on the recent papers in your reference list. Electronic journals tend to be quicker to publication, and probably also to acceptance, than traditional ones, of course, so (ii) will be biased in that direction to some extent.
(iv) Finally I have not read the introduction to your paper but often introductions are crucial in 'selling' a paper to the referee. If you say what the paper sets out to do, clearly and concisely, then your chances of getting a referee's report more quickly, and one which will be positive, will increase.
(v) The paper was submitted to Arxiv last November. That means you have a certain distance from it. Do a critical health check on it from the point of view of wording, sentence structure, spelling, etc., before sending it to any journal. Get a friend or colleague to do a quick read of intros etc. so as to get a second opinion on wording. Think of the poor referee, make their job easier. They need to be able to evaluate the paper fairly quickly. Check, yourself, for typos (and, of course, don't trust spell checking programs on this). My reaction to writing a report when there are clearly lots of typos is to put it off for a few weeks... therefore longer turn around time.
Finally I feel that your should have put the question in a different form. Specific journal suggestions are hardly the responsibility of MO but your general point on how to select journals for such a paper is a valid one .